Rocket equation (Score 3)

Ouch .. this hurts! I am sorry dudes but I just can't get my head around this. Reason:
I begin with the classical (ie. non-relativistic) rocket equation (I use the classical version because relativistic effects only become important for exhaust velocities greater than about 95% the speed
of light, which is not the case for the powers and speeds we are talking about here).

The rocket equation is:

dv = u ln [ ( M + m ) / M ]

where:

dv = change in ship velocity
u = exhaust velocity
M = ship mass, without including reaction mass
m = reaction mass ejected from ship

Now in general, to get from one place to another a ship must accelerate for some time T /2, then coast at top velocity for a time , then decelerate over a time T /2.
The total change in velocity is v, but since the ship speeds up and slows back down to rest, the maximum velocity is v /2. The total trip time is the time spent accelerating/decelerating plus the time spent coasting.
Now the power required to eject the reaction mass at the given exhaust velocity is equal to the rate of change of kinetic energy of the reaction mass, which is half the mass-loss rate dm /dt times the square of the exhaust velocity.

And that's that!